“Michiel Doorman was born on 1 october 1962 in Eindhoven, the Netherlands. He completed his secondary education in 1981 at the Minkema College in Woerden. In1988 the Utrecht University awarded him a masters in Mathematics for his thesis on the extension of a proposition in intuitionistic logic for automated theorem proving. He minored in Computer Science. From 1988 he has been working at the Freudenthal Institute. Until 1992 he was mainly devoted to software development. During the following years he has been involved in curriculum and teacher training projects, mainly concerning the role of information and communication technology in mathematics education. Since 1994, this work concentrated on upper secondary (pre-university) mathematics education in a research project on the integration of the graphing calculator, in a curriculum development project (Profi), and in a project that aimed at guiding the Biology, Chemistry, Physics and Mathematics departments in schools to cooperate. In 1998 he started his PhD research study.” [my emphasis]

this is the proceeding pdf: “What Can Mathematics Education Contribute To Preparing Students For Our Future Society?“.

Michiel Doorman fourth from left (Source: ICRIEMS website 2016)

Why does Doorman in 2016 claim success for RME while it failed ?

Around 2005 there was much discussion in Holland – a real math war – about arithmetic in elementary schools. The academy of sciences (KNAW) set up a committee to look into this.

Recall the graphical display of the math war between RME and TME, and the solution approach of NME. These abbreviations are:

RME = realistic mathematics education

TME = traditional mathematics education

NME = neoclassical mathematics education

(i) This KNAW-committee concluded in 2009 (see the English summary on page 10 in the report) that pupil test scores for RME and TME did not really differ. Paraphrased: one cannot claim a special result for RME. Observe that many test questions contained contexts.

“Growing concern about Dutch children’s mathematical proficiency has led in recent years to a public debate about the way mathematics is taught in the Netherlands. There are two opposing camps: those who advocate teaching mathematics in the “traditional” manner, and those who support “realistic” mathematics education. The debate has had a polarizing effect and appears to have little basis in scholarly research.” [This neglects my third position with NME.]

“The public debate exaggerates the differences between the traditional and realistic approaches to mathematics teaching. It also focuses erroneously on a supposed difference in the effect of the two instructional approaches whereas in fact, no convincing difference has been shown to exist.” (KNAW 2009)

(ii) Doorman in Yogyakarta 2016 is unrepentingly for RME. He refers to key authors on RME, and takes a question of TIMMS 2003 with an international score of 38% and a Dutch score of 74% and claims, misleadingly:

“This cannot fully [be] attributed to the implementation of RME, but it strengths [sic] the feeling that this approach contributes to the quality of mathematics education.” [my emphasis]

(iii) Subsequently, I criticised the KNAW report on these counts, and neither KNAW nor Michiel Doorman have responded to this criticism:

Before the report was published by alerting the committee chairman to Elegance with Substance (2009, 2015), that however is not included in the references.

In 2014 explicitly for the collective breach of research integrity, for either neglecting or maltreating my books Elegance with Substance (2009, 2015) and Conquest of the Plane (2011) and Dutch Een kind wil aardige en geen gemene getallen (2012) notably on issues of arithmetic (the present subject): the pronunciation of numbers and notation of mixed numbers.

In 2015 for neglecting the issue that TME prepares for algebra while RME doesn’t. The KNAW report uses the outcome of test questions and not the intermediate steps. Pupils who can only use RME will be very handicapped for algebra in secondary education.

See my 2016 letter and its supplement to the president of KNAW and director of CPB about the failure of the KNAW report and the neglect of criticism.

A repeat exercise that isn’t quite superfluous

I have explained, to boring repetition, that the Freudenthal Head in the Clouds Realistic Mathematics Institute (FHCRMI) should not be at a university. Please observe that first there was criticism on the failure of “realistic mathematics education” (RME) and only later it was discovered that Hans Freudenthal had actually abused the work by Pierre van Hiele. There also is a sound scientific explanation why it is a failure: namely a confusion of processes of learning with applied mathematics.

There is little advantage in repeating this analysis,
neither for each and every individual working at FHCRMI.

For example, stating that Michiel Doorman works at FHCRMI should be sufficient. That he is at FHCRMI does not imply that he can indeed be at university and that his “thesis” and “PhD title” are proper.

However, the following points cause that it isn’t quite superfluous to look into Doorman’s qualifications and work.

Michiel Doorman is member of the board of NVvW, the Dutch association of mathematics teachers. See my recent letter with a Red Card for this board. Thus it helps for the next annual meeting of NVvW in November 2016 to be specific.

Also, there is my letter of April 15 2016 to NRO, the Dutch organisation for the distribution of funds for research in mathematics education. I advise them to stop subsidising FHCRMI. It so happens that Michiel Doorman did a project ODB08008 for them in 2009-2012 on the “digital mathematics environment” and “efficient exercising mathematics” (DWO). It will be helpful for NRO to see that, for example, Doorman is an ideologue and no scientific researcher. This is related to the following.

There is the new impulse for “21st century skills” or in Holland “Onderwijs2032“. Part of the attention is for soft skills, part of the attention is for computer programming, part is elsewhere. ICT brings us to the work of Doorman too. There has already been a major disaster with the neglect of computer algebra since 1990. For example DWO at FHCRMI tends to present many Java applets that lack the flexibility of computer algebra. Don’t think that these issues are easy to resolve, but I do hold that the decisions have been driven by ideology and that the results are a disaster and a great waste of funds: penny wise pound foolish. See for example these two reports by the Inspectorate for Education: In 2002, mathematical topologist Hans Freudenthal is called a “pedagogue” while he had no education or training on this, and they assume that FHCRMI knows about ICT while the report doesn’t mention computer algebra but applets on “wisweb”. In 2006, the “waarderingskader” (inspection standards) doesn’t seem to realise that computer algebra can used in all subjects that use mathematics.

Above, I mentioned four neutral points. Following Sherlock “Google” Holmes I already debunked the event in Yogyakarta. Let us look at the other three points. Beware of confusion.

Ad 1. Doorman’s “thesis” of 2005

In the “thesis”, Doorman basically refers to Paul Drijvers at FHCRMI for computer algebra, but Drijvers is no light on this either. For the subject of the thesis (“Furthermore, it has been examined what role computer tools could play in learning mathematics and physics.”) it would have made much sense to look deeper into computer algebra.

Also check my analysis that Koeno Gravemeijer is no scientist but an ideologue for “realistic mathematics education” (RME) who misrepresents issues on “21st century skills” (in Holland “Onderwijs2032“), and who doesn’t see the revolution by computer algebra. (Dutch readers can look here too.)

On p58-59 Doorman critically adopts RME, and remember that this was in 2005, while RME was under discussion, see (a) the discussion in 2006 between Robbert Dijkgraaf (who has no qualification for math ed at this level) en Kees Hoogland (a RME ideologue, see below), which report was written by RME supporter Martinus van Hoorn, and (b) while Jan van de Craats (who has no qualification for math ed at this level) was protesting about RME, and published “Waarom Daan en Sanne niet kunnen rekenen” in 2007. See my criticism w.r.t. Jan van de Craats (fighting his math war on the side of TME and neglecting NME since 2008).

Doorman refers to Freudenthal for “guided reinvention”, but this is a wrong reference (he may only have coined the phrase but not the concept), and Doorman’s thesis does not refer to the true inventor of the concept (guide through levels of insight) Pierre van Hiele at all.

I will not look at this “thesis” in detail because there is really no reason to so so now.

Ad 2. Curriculum vitae

Doorman’s cv shows that he got a mathematics degree and continued at Freudenthal Head in the Clouds Realistic Mathematics Institute (FHCRMI), thus without a teaching degree in mathematics and without proper training in research of mathematics education.

The KNAW report of 2009 showed that FHCRMI doesn’t do research on arithmetic education, and one should not suppose that this is different for other areas.

Thus Michiel Doorman is neither a teacher of mathematics nor a researcher in the education of mathematics.

We find no qualification for teaching and research, but immersion into ideology, and while he is involved in programming and the role of ICT for (mathematics) education, there are only perfunctory statements on computer algebra (for all subjects that use mathematics).

Ad 3. Webpage

Observe that “context-based mathematics education” is a rephrasing of “realistic mathematics education” (RME). Also “inquiry based learning” is basically RME, with contexts as the starting point for the “inquiry” (constrained by learning goals).

Someone really interested in didactics and RME would also have been interested in my analysis that shows that RME is a confusion and an ideology.

Observe also that the sciences are easy victims of RME. The sciences do not care much about mathematics education, and when RME people flock in to assist in the learning of the sciences, and when student learning time for mathematics is actually spent on the sciences, then professors of physics or biology might hardly object. For RME the sciences are useful window dressing, since those provide both contexts and an aura of respect and acceptance, and an argument that “students are learning something” (even if it isn’t mathematics). There is a curious historical link-up of mathematics with the beta sciences while there are also the humanities (alpha) and social sciences (gamma), see here. But is it really curious, and isn’t there a method, that the human and social scientists who know the techniques and who also do research by observation are kept out from this association between “mathematics education” and “science education” ?

Possible confusions that are triggered immediately

Stating the above might immediately trigger some confusions.

As member of the NVvW board Doorman might argue “not to look into the criticism on FHCRMI because of an interest there”. Instead, he should rather take the initiative and make sure that this criticism was answered in decent manner rather than burked. If his interest is so large that he cannot speak freely on science then he should rather not be in the board.

Doorman in a 2015 text in Euclides, the journal of NVvW, referred to (intellectually stealing) Freudenthal and not to (victim) Van Hiele. When asked to correct, he didn’t reply to this very question (see page 8) but talked around it, see my deconstruction of his “reply”. Potentially Doorman just didn’t have the relevant knowledge about didactics, for his “thesis” refers to RME but not to Van Hiele. If Doorman had looked into this criticism of mine, he could have been a bridge of understanding for the other members of the NVvW board and readers of Euclides, but he wasn’t.
PM 1. I don’t understand either why these people didn’t see that he dodged the question.PM 2. Doorman in Indonesia sheets 44-49 on RME repeats the reference on anti-didactic inversion to (intellectually abusing) Freudenthal at the cost of (victim) Van Hiele. Thus I asked a correction, he dodged the question, and proceeds as if there would be no problem (and likely not informing the audience about the criticism).

Thus Doorman is neither teacher nor researcher: so what is he doing in the board of NVvW ? From qualifications, actually their lack, and work, actually the lack of answers, it is a small step towards wondering about motivation. A good hypothesis is: he is spreading the gospel of RME and blocking criticism. If Doorman wants to become professor at FHCRMI he must show that he is a true sectarian of RME. He is polishing up his cv and can now claim that he has been involved in the community of teachers, even when it was dodging questions. This is a hypothesis only, and one might also offer other explanations like blindness.

I wonder who paid for this trip to Yogyakarta. It is also known that Freudenthal Head in the Clouds Realistic Mathematics Institute (FHCRMI) still is busy in establishing footholds over the world even though RME has failed. See also FIUS.org, who apparently neglect the KNAW report or my criticism.

Yes, there is also NRO-project supervisor Frits Beukers, but he is professor of mathematics also without qualification for mathematics education research. Beukers presently is chairman of the Platform Wiskunde Nederland (PWN)-education committee, but would represent the universities since he has no qualification for primary or secondary education or the trade colleges. In that committee we also see Kees Hoogland, who abused the biography by John Allen Paulos for RME, and who has not corrected yet and who refuses to give an English translation of the key section.

Disclaimer: Can I be unbiased ?

I stated that I would give an unbiased assessment. Can I really do so ? Undoubtedly the reader will make up one’s own mind, but my perception is that I have been fair and unbiased.

Doorman’s “thesis” of 2005 is closely related to the education on the derivative. There was ample scope for a meeting of minds. When Doorman sticks with RME and Java applets, instead of NME and computer algebra, it is all of his own choosing. The differences in positions can be mentioned:

Check my proposal since 2007 for an algebraic approach to the derivative, see e.g. Conquest of the Plane (2011).

COTP was also programmed in Mathematica – a system for doing mathematics on the computer (a.k.a. “digital environment”) .

Also, I am a user of computer algebra since 1993, while Doorman tends to use other programs that don’t have the flexibility of computer algebra.

Doorman was editor of TD-Beta when I submitted a short note in 2012 on my invention of the algebraic approach to the derivative. This was maltreated. See here anonymised and see here with full names (it is a scientific discourse and no private exchange).

At the NVvW annual convention of 2015 when Doorman was elected to the board, I had my doubts but opted still mildly optimistically for the benefit of the doubt. I had no experience with this TD-Beta journal and perhaps everything was an unfortunate misunderstanding. It doesn’t happen so very often that someone can propose a new approach to the education on the derivative. The performance of last eight months however gives evidence of the mindset of an ideologue.

Conclusion: Doors of perception

Any link to Doorman’s name is coincidental, and it is also a coincidence that the Dutch family name “Doorman” indeed is related to the English “doorman” (at least according to the Meertens institute).

The phrase “doors of perception” comes to mind when looking at Doorman’s presentation sheet “Aim of Primas” that states:

“A question not asked is a door not opened!”

This implies that an opened door is a result of a question. (This need not be an open door.)

This doesn’t imply that asking a question will open a door. (The question might e.g. be ignored.)

The message of this present weblog is, amongst others, that there are some crucial questions that Michiel Doorman refuses to look into and apparently doesn’t want to answer. He is employed at the Freudenthal Head in the Clouds Realistic Mathematics Institute, that should not be at a university. Apparently Doorman did not inform his hosts in Indonesia about the existing critique either.

The earlier discussion on Stellan Ohlsson brought up the issue of abstraction. It appears useful to say a bit more on terminology.

An unfortunate confusion at wikipedia

Wikipedia – no source but a portal – on abstraction creates a confusion:

Correct is:“Conceptual abstractions may be formed by filtering the information content of a concept or an observable phenomenon, selecting only the aspects which are relevant for a particular purpose.” Thus there is a distinction between abstract and concrete.

Confused is: “For example, abstracting a leather soccer ball to the more general idea of a ball selects only the information on general ball attributes and behavior, eliminating the other characteristics of that particular ball.” However, the distinction between abstract and concrete is something else than the distinction between general and particular.

Hopelessly confused is:“Abstraction involves induction of ideas or the synthesis of particular facts into one general theory about something. (…) Bacon used and promoted induction as an abstraction tool, and it countered the ancient deductive-thinking approach that had dominated the intellectual world since the times of Greek philosophers like Thales, Anaximander, and Aristotle.” This is hopelessly confused since abstraction and generalisation (with possible induction) are quite different. (And please correct for what Bacon suggested.)

A way to resolve such confusion is to put the categories in a table and look for examples for the separate cells. This is done in the table below.

In the last row, the football itself would be a particular object, but the first statement refers to the abstract notion of roundness. Mathematically only an abstract circle can be abstractly round, but the statement is not fully mathematical. To make the statement concrete, we can refer to statistical measurements, like the FIFA standards.

The general statement All people are mortal comes with the particular Socrates is mortal. One can make the issue more concrete by referring to say the people currently alive. When Larry Page would succeed in transferring his mind onto the Google supercomputer network, we may start a philosophical or legal discussion whether he still lives. Mutatis mutandis for Vladimir Putin, who seems to hope that his collaboration with China will give him access to the Chinese supercomputers.

Category (mistake)

Abstract

Concrete

General

The general theory of relativity

All people living on Earth in 2015 are mortal

Particular

The football that I hold is round

The football satisfies FIFA standards

The complex relation between abstract and general

The former table obscures that the relation between abstract and general still causes some questions. Science (Σ) and philosophy (Φ) strive to find universal theories – indeed, a new word in this discussion. Science also strives to get the facts right, which means focusing on details. However, such details basically relate to those universals.

The following table looks at theories (Θ) only. The labels in the cells are used in the subsequent discussion.

The suggestion is that general theories tend to move into the abstract direction, so that they become universal by (abstract) definition. Thus universal is another word for abstract definition.

A definition can be nonsensical, but Σ strives to eliminate the nonsense, and officially Φ has the same objective. A sensible definition can be relevant or not, depending upon your modeling target.

Let us redo some of the definitions that we hoped to see at wikipedia but didn’t find there.

Abstraction is to leave out elements. Abstractions may be developed as models for the relevant branch of science. The Van Hiele levels of insight show how understanding can grow.

A general theory applies to more cases, and intends to enumerate them. Albert Einstein distinguished the special and the general theory of relativity. Inspired by this approach, John Maynard Keynes‘s General Theory provides an umbrella for classical equilibrium (theory of clearing markets) and expectational equilibrium (confirmation of expectations doesn’t generate information for change, causing the question of dynamic stability). This General Theory does not integrate the two cases, but merely distinguishes statics and its comparative statics from dynamics as different approaches to discuss economic developments.

Abstraction (A) is clearly different from enumeration (G). It is not impossible that the enumeration concerns items that are abstract themselves again. But it suffices to assume that this need not be the case. A general theory may concern the enumeration of many particular cases. It would be statistics (GΣ) to collect all these cases, and there arises the problem of induction (GΦ) whether all swans indeed will be white.

Having both A and G causes the question how they relate to each other. This question is studied by R.

This used to be discussed by traditional epistemology (RΦ(a)). An example is Aristotle. If I understand Aristotle correctly, he used the term physics for the issues of observations (GΣ) and metaphysics for theory (AΦ & GΦ). I presume that Aristotle was not quite unaware of the special status of AΣ, but I don’t know whether he said anything on this.

Some RΦ(a) neglect Σ and only look at the relation between GΦ and AΦ. It is the price of specialisation.

Specialisation in focus is also by statistical testing (RΣ(b)) that only looks at statistical formulations of general theories (GΣ).

The falsification theory by Karl Popper may be seen as a philosophical translation (RΦ(b)) of this statistical approach (RΣ(b)). Only those theories can receive Popper’s label “scientific” that are formulated in such manner that they can be falsified. A black swan will negate the theory that all swans are white. (1) One of Popper’s problems is the issue of measurement error, encountered in RΣ(b), with the question how one is to determine sample size and level of confidence. Philosophy may only be relevant if it becomes statistics again. (2) A second problem for Popper is that AΣ is commonly seen as scientific, and that only their relevance can be falsified. Conservation of energy might be relevant for Keynes’s theory, but not necessarily conversely.

The Definition & Reality methodology consists of theory (RΦ(c)) and practice (RΣ(c)). The practice is that scientists strive to move from the particular to AΣ. The theory is why and how. A possible intermediate stage is G but at times direct abstraction from concreteness might work too. See the discussion on Stellan Ohlsson again.

Conclusions

Apparently there exist some confusing notions about abstraction. These can however be clarified, see the above.

The Van Hiele theory of levels of insight is a major way to understand how abstraction works.

Paradoxically, his theory is maltreated by some researchers who don’t understand how abstraction works. It might be that they first must appreciate the theory before they can appreciate it.

To my surprise, today gives more on psychology. Since highschool I denote this as Ψ. I appreciate social Ψ (paper 1996) but am not attracted to other flavours of Ψ.

Last week we looked at some (neuro-) Ψ on number sense, and a few days ago at some cognitive Ψ. Dutch readers may look at some comments last year w.r.t. the work by Leiden Ψmetrist Marian Hickendorff who explains that she is no expert on math education but still presents research on it.

Today I will look at what Dutch Ψist and education researcher Ben Wilbrink states about the work by math education researcher Pierre van Hiele (1909-2010). I already observed a few days ago that Wilbrink didn’t understand Van Hiele’s theory of levels of insight. Let me become more specific.

ME and MER are a mess, but Ψ maybe too

The overall context is that math education (ME) and its research (MER) are a mess. Mathematicians are trained for abstraction and cannot deal well with real existing pupils and the empirical science of MER.

When Ψ has criticism on this, it will be easy for them to be right.

Unfortunately, Ψ appears to suffer from its own handicap. Ψ people namely study Ψ. They do not study ME or MER. Ψists invent their own world full of Ψ theories alpha to omega, but it is not guaranteed that this really concerns ME and MER. We saw this in (neuro-) Ψ and in cognitive Ψ in above weblog texts. It appears also to hold for Wilbrink. Whether Ψ is a mess I cannot judge though, since I am no Ψist myself.

Ψ itself has theories about how people can be shortsighted. But we don’t need such theory. A main element in the explanation is that Ψists tend to regard mathematicians as the experts in ME, while those are actually quite misguided. A mathematician’s view on ME tends put the horse behind the carriage. Then Ψ comes around to advise ways to do this more efficiently.

This has become an issue of research integrity

I have asked Ben Wilbrink to correct some misrepresentations. He refuses.

He might have excellent reasons for this. My problem is that he doesn’t state them. I can only guess. One potential argument by Wilbrink is that he does Ψ. Perhaps he means to say that when I would get my third degree in Ψ too then I might better understand his misrepresentations. This is unconvincing. A misrepresentation remains a misrepresentation, whatever the amount of Ψ you put into it. Unless Wilbrink means to say that Ψ is misrepresentation by itself. Perhaps.

But: Wilbrink’s refusal to provide answers to some questions turns this into an issue in research integrity.

Wilbrink (1944, now 70+) originally worked on the Ψ approach to test methodology (testing people rather than eggs). See for example the Item Response theory by Arpad Elo and Georg Rasch, also discussed in my book Voting Theory for Democracy. The debate in Holland on dismal education in arithmetic causes Wilbrink to emphasize the (neglected) role of Ψ. He also tracks other aspects, e.g. his website lists my book Elegance with Substance(EWS) (2009), but he makes his own selection. Perhaps he hasn’t read EWS. At least he doesn’t mention my advice to a parliamentarian enquiry into mathematics education. All this is fine with me, and I appreciate much of Wilbrink’s discussions.

However, now there is this issue on research integrity.

Let us look at the details. The basic evidence is given by Wilbrink’s webpage (2012) on Pierre and Dina van Hiele-Geldof (retrieved today).

1. Having a hammer turns everything into a nail (empirics)

If you want to say something scientifically about mathematics education (ME), then you enter mathematics education research (MER).

When you meet with criticism by people in MER that you overlook some known results, then check this.

Ben Wilbrink overlooks some known results.

But he refuses to check those, even when asked to.

In particular, he states that the Van Hiele theory of levels of insight would not be empirical.

But my books and weblog texts, also this recent one, explain that it is an empirical theory. I informed him about this. Wilbrink must check this, ask questions when he doesn’t understand this, and give a counterargument if he does not agree. But he doesn’t do that. What he does, is neglect MER, and simply state his view, and neglect this criticism. Thus:

As a scientist Wilbrink should give a counterargument, but he merely neglects it.

3. Having a hammer turns everything into a nail (Freudenthal)

A third case that Wilbrink (here, w.r.t. p233 ftnt 38 again) shows that he doesn’t understand the subject he is writing about, is that he lumps Van Hiele and Freudenthal together, i.e. on the theory of levels. But their approaches are quite different. Van Hiele has concrete versus abstract, Freudenthal has pure versus applied mathematics. Freudenthal’s conceptual error is not to see that you already must master mathematics before you can do applied mathematics. You will not master mathematics by applying it when you cannot apply it yet. Guided reinvention is a wonderful word, like sim sala bim.

It is a huge error by Wilbrink to not see this distinction. Wilbrink doesn’t know enough about MER. This turns from sloppy science into an issue of research integrity when he does not respond to criticism on this.

Remarkably, Wilbrink (here, on Structure and Insight) rightly concludes that Van Hiele is critical of Freudenthal and doesn’t actually belong to that approach. Apparently, it doesn’t really register. Wilbrink maintains two conflicting notions in his mind, and doesn’t care. (See also points 10 and 14 below.)

4. Having a hammer turns everything into a nail (Kant)

Wilbrink looks at ME and MER from the angle of Ψ. This looks like a valuable contribution. He however appears to hold that only Ψ is valid, and MER would only be useful when it satisfies norms and results established by Ψ. This is scientifically unwarranted.

There are cases in which Ψ missed insights from MER. See above. I have noted no Ψist making the observations that can be found in Elegance with Substance.

The Van Hiele theory is a general theory in epistemology (see here), and thus also Ψ must respect that. When Wilbrink doesn’t do that, he should give an argument.

A conceivable argument by Wilbrink might be that Van Hiele did not publish a paper in a journal on philosophy (my notation Φ) so that the sons and daughters of Kant could have hailed it as a breakthrough in epistemology. The lack of this seal of approval might be construed as an argument that Ψ and Wilbrink would be justified to neglect it. This would be an invalid argument. When Wilbrink studies MER and Van Hiele’s theory of levels, and reads about Van Hiele’s claim of general epistemological relevance, then every academic worth his or her salt on scientific methodolgy, and especially Ψists, can recognise it for what it is: a breakthrough.

5. Having a hammer turns everything into a nail (testing validity)

Wilbrink’s question whether there has been any testing on validity on Van Hiele’s theory at first seems like a proper question from a Ψist, but neglects the epistemological status of the theory. He would require from physicists that they “test” the law of conservation of energy, or from economists that they “test” that savings are what remain from income after consumption. This is quite silly, and only shows that Wilbrink did not get it. Perhaps his annoyance about Freudenthal caused him to attack Van Hiele as well ? Wilbrink should correct his misrepresentation, or provide a good reason why being silly is good Ψ.

6. Having a hammer makes you require that everyone is hammering

Wilbrink suggests that Pierre and Dina Van Hiele – Geldof performed “folk psychology”. This runs counter to the fact that Pierre studied Piaget, and explicitly rejected Piaget’s theory of stages. His 1957 thesis (almost 60 years ago) has three pages of references that include also other Ψ. Perhaps Wilbrink requires that they should have studied more of Ψ. That might be proper when the objective was to become a Ψist. But the objective was to do MER. Dina did the thesis with Langeveld, a pedagogue, and Pierre with Freudenthal, mathematician and not known yet for the educational theories that he stole from Pierre (and distorted, but it remains stealing).

If the Ψists would succeed in presenting a general coherent and empirically corroborated theory, that every academic can master in say a year, then perhaps Ψists might complain that this is being neglected. Now that Ψists however create a wealth of different approaches, then researchers in MER are justified in selecting what is relevant for their subject, and proceed with the subject.

Wilbrink’s suggestion on “folk psychology” is disrespectful and slanderous.

7. Having a hammer makes you look for nails at low tide (pettifoggery)

Wilbrink reports that Dina van Geldof mentions only the acquisition of insight and does not refer to the relevance of geometry for a later career in society. Perhaps she doesn’t. Her topic of study was acquisition of insight. Perhaps Wilbrink only makes a factual observation. What is the relevance of this ? It is a comment like: “Dollar bills don’t state that people also use them in Mexico.” Since Wilbrink reports this in the context of above disrespectful “folk psychology”, the comment only serves to downgrade the competence of Dina van Geldof, and thus is slanderous. As if she would not understand it, when Pierre explained to her that his theory of levels had general epistemological value.

8. Having a hammer makes you look for nails in 1957

Wilbrink imposes norms of modern study design and citation upon the work of the Van Hieles in 1957 (when Pierre was 48). The few references in Pierre’s “Begrip en inzicht” (2nd book, not the thesis, also translated as “Stucture and insight”) cause Wilbrink to hold, in paraphrase,

“by not referring, Van Hiele reduces his comments to personal wisdoms, by which he inadvertedly downgrades them.”

This is a serious misrepresentation, even though the statement is that Van Hiele’s texts were more than just personal wisdoms.

(a) It is true that Van Hiele isn’t the modern researcher who always refers and is explicit about framework and study design. What a surprise. The observation is correct that norms of presentation of results have changed. Perhaps authors in the USA 1957 already referred, but this need not have been the case in Europe. (See a discussion on this w.r.t. John Maynard Keynes.)

(b) The suggestion as if Van Hiele should have referred is false however. In that period the number of researchers and size of literature were relatively small, and an author could assume that readers would know what one was writing about. Some found it also pedantic to include footnotes.

Thus: (i) The lack of footnotes does not in any way reduce Van Hiele’s comments to “personal wisdoms”. Wilbrink is lazy and if he is serious about the issue then he should reconstruct the general state of knowledge in that period. (ii) The comment must be rewritten in what is factually correct, and the insinuation must be removed.

9. Having a hammer makes you put nails in other people’s mouths

Wilbrink refers to an issue on fractions. He quotes Van Hiele’s suggestion to use tables of proportions, which has been adopted by the Freudenthal Institute, and quotes criticism by modern mathematicians Kaenders & Landsman that those tables block insight into algebra.

This is a misrepresentation.

This is an example of that a Ψist quotes mathematicians as authorities, and regards their misunderstanding as infallible evidence. A student of MER however would (hopefully) see that there is more to it.

The very quote by Van Hiele contains his suggestion to look at multiplication. Indeed, the book “Begrip en Inzicht” chapter 22 contains a proposal to abolish fractions, and to deal with that algebraically – what Kaenders & Landsman may not know about.

The true criticism is that the Freudenthal Head in the Cloud Realistic Mathematics Institute mishandled Van Hiele’s work: (a) selected only an easy part, and (b) did not further develop Van Hiele’s real approach.

A proposal how Van Hiele’s real approach can be developed is here. I agree with Kaenders & Landsman to the extent that presenting only such tables is wrong, and that also the algebraic relation should be specified. The student then has the option to use either, and learn the shift.

Curiously, Wilbrink comments on this chapter 22 with some approval. Thus he should have seen that he provided a false link between Van Hiele on tables of proportion and the critique by Kaenders & Landsman.

10. Having a hammer makes you hate who refuses to be a nail

“Many original ideas can be found in this book. I came upon them in analyzing dubious theories of both psychologists and pedagogues. It is not difficult to unmask such theories: simply test them in practice. Often this is not done because of the prestige of the theory’s proponents.”

Wilbrink’s judgement (my translation):

“The quoted opinion is incredibly arrogant, lousy, or how do you call such a thing. Van Hiele is mathematician, and makes the same error here as Freudenthal made in his whole later life: judging the development of psychological theory not in the context of psychology, but in the context of one’s own common sense. This clearly gives gibberish. Thus I will continue reading Van Hiele with extraordinary suspicion.”

My comments on Wilbrink:

Van Hiele was a mathematician but also a teacher, with much attention for the empirics of education. This is quite in contrast with Freudenthal who lived by abstraction. (Freudenthal did not create a professorship in math education for Van Hiele, but took the task himself.)

Van Hiele does precisely what Wilbrink requires: look at Ψ and look at empirics (in this case: practice). The only thing what happens is that Van Hiele then rejects Ψ, and this is what Wilbrink doesn’t swallow. While Van Hiele does MER, Wilbrink redefines this as Ψ, and then sends Van Hiele to the gallows for not sticking to some Ψ paradigm.

It is useful to mention that Van Hiele does the same thing in the preface of his thesis. He states that Ψ theories have been shown inadequate (his references are three pages) and that he will concentrate on the notion of insight as it is used in educational practice. He opposes insight to rote learning, and mentions the criterion of being able to deal with new situations that differ from the learning phase.

It is incorrect of Wilbrink to distinguish only the categories of either Ψ or “one’s own common sense” or “folk psychology”. It is quite obvious why Van Hiele cannot find in books on Ψ what he is looking for and actually does: He presents his epistemological theory of levels. Those aren’t in those books on Ψ. If Van Hiele would do what Wilbrink requires, then he cannot present his theory of levels, since Wilbrink’s strict requirements would force him to keep on barking up the wrong tree. It beats me why Wilbrink doesn’t see that.

11. Having a hammer turns your foot into a nail

Wilbrink also quotes from viii:

“Some psychologies lay much stress on the learning of facts. The learning of structures, however, is a superior goal. Facts very often become outmoded; they sink into oblivion because of their lack of coherence. In a structure facts have sense; if part of a structure is forgotten, the remaining part facilitates recall of the lost one. It is worth studying the way structures work because of their importance for the process of thinking. For this reason a considerable part of my book is devoted to structures.”

Wilbrink’s comment on this is (my translation):

“For me this is psychological gibberish, though I rather get what Van Hiele intends (…)”

By which it is established that Wilbrink understands gibberish and may call gibberish what he understands.

12. Having a hammer makes that you run in a loop of nails

Wilbrink’s subsequent quote from Structure and Insight:

“In this book you will find a description of a theory of cognitive levels. I show you how levels of thinking demonstrate themselves, how they come into existence, how they are experienced by teachers and how by pupils. You will also see how we can take account of those levels in writing textbooks.”

Wilbrink (my translation):

“You cannot simply do this. At least Van Hiele must show by experiment that intersubjective agreement can be reached about who when what level has demonstrated by operational achievements (because we cannot observe thoughts directly). (…) Indeed, at least for himself it is evident. Can this idea be transferred to others ? Undoubtedly, for other people have invited him to make this English translation of his earlier book. But that is not the point. The crucial point is: does his theory survive empirical testing?”

My comment: It is a repetition of the above, but it shows that Van Hiele’s repeated explanation about the epistemological relevance of his theory for educational practice continues, time and time again, to elude Wilbrink’s frame of mind.

Of course, statistical science already established before 1957 that the golden standard of experimental testing consists of the double blind randomized trial. Instead, Van Hiele developed his theory over the course of years as teacher in practice. Though he mentions didactic observations already from his time as a student in highschool. But we are back in a repetitive loop when we must observe that it is false to require statistics for Van Hiele’s purposes.

13. Having a hammer makes you avoid number 13 for fear that it might make you superstitious

Hermann von Helmholtz, on the law of conservation of energy (source: wikimedia commons)

14. Having a hammer makes you miss a real nail

Wilbrink (2012) refers to the MORE study of 1993 that defined realistic mathematics education (RME) as consisting of:

It is actually nice that Van Hiele is mentioned in 1993, for at least since 2008 he isn’t mentioned in the Freudenthal Head in the Clouds Realistic Mathematics Institutewiki on RME (retrieved today). His levels have been replaced by Adri Treffer’s concept of “vertical mathematization”. Wilbrink might be happy that he doesn’t have to criticise the levels at FHCRMI anymore. It is now a vague mist that eludes criticism.

Wilbrink’s criticism of Freudenthal’s didactic phenomenology and Wiskobas are on target. It is indeed rather shocking that policy makers and the world of mathematics teaching went along with the nonsense and ideology. The only explanation is that mathematicians made a chaos with their New Math. If Pierre van Hiele had been treated in scientific decent fashion, his approach would have won, but Freudenthal was in a position to prevent that.

Wilbrink apparently thinks that Van Hiele belongs to the Freudenthal group, even though he observes elsewhere that Van Hiele rejects this. Wilbrink assumes both options, and his mind is in chaos.

Wilbrink doesn’t see that the Freudenthal clique only mentions Van Hiele to piggyback on his success, to manoeuvre him out, and later create some matching phrases so that Van Hiele doesn’t have to be mentioned anymore.

The following is a repetition of point 5, but it can be found on this particular page & section by Wilbrink, and may deserve a comment too. Namely, regarding Van Hiele as a pillar of realistic mathematics education, Wilbrink states (my translation):

“Okay, I can infer that the theory of levels can be found in Van Hiele’s thesis, but that thesis is of a conceptual nature, and it doesn’t contain empirical research. Van Hiele doesn’t deny the latter, see the passage on his pages 188-189; but that is really rather sensational: everyone parrots his theory of levels, without looking for empirical support. Every well-thinking person, who has read his Popper for example, can see that you can do just anything with that ‘theory of levels’: It is in the formulation by Van Hiele 1958 [article following the 1957 thesis ?] a theory that excludes almost nothing. I return to this extensively on the Van Hiele page.”

My comments for completeness:

Van Hiele’s theory is as empirical as the law of conservation of energy or the economic principle that savings are the remainder of income after consumption. This is not pure mathematics but it applies to reality. Thus Van Hiele’s theory is hugely empirical. See the former weblog text.

Van Hiele’s thesis p188-189 indeed mentions the subsequent relevance of statistical testing to ground out details. This is something else than testing on falsification. What Van Hiele states is not quite what Wilbrink suggests. The fact of the lack of statistical testing is correct. But Van Hiele does not subscribe to Wilbrink’s criterion of “empiricism”.

Van Hiele does not expect that there will be much statistical development of the levels. Therefor he judges that his theory will tend to be of more value for teachers in practical teaching.

You can do with the theory of levels as much as with the law of conservation of energy. A bit, but a crucial bit. Who has read Popper will see that the idea of falsification must make an amendment on definitions.

Thus, if Wilbrink had had an open mind on epistemology, he could have nailed the FHCRMI for producing nonsense and abusing the wonderful theory by Van Hiele. He missed.

But the key point is that his also misinforms his readership, and refuses to correct after he has been informed about it.

15. Having a hammer makes that only masochist nails like you

Wilbrink’s discussion of Van Hiele’s thesis chapter 1 (here, “Wat is inzicht?”) shows a lack of understanding about the difference between a theorem and a proof. Euclid turns in his grave.

Wilbrink makes a distinction between “mathematics and psychology of mathematics”, without explanation or definition, perhaps in the mood of writing for Ψists who will immediately smell the nest and cheer and be happy.

Wilbrink writes “Brrrrr” (check the r’s) when Van Hiele distinguishes insight based upon inference and insight based upon non-inference. Wilbrink does not explain whether his Brrrrr is based upon inference or non-inference.

Wilbrink fears that Van Hiele will base his didactic insight upon “reason” instead of “theory with empirical testing”. He does not explain what is against reasoning and teaching experience and reading in the literature, for developing a new theory. Perhaps Wilbrink thinks that true theories can only be found in books of Ψ ?

Wilbrink’s final judgement on Van Hiele’s thesis chapter 1 is that it is a “tattle tale”. It is a free world, and Wilbrink may think so and put this on his website. But if he wants to be seen as a scientist, then he should provide evidence. In this case, Van Hiele clearly stated that he found the Ψ theories useless, so that he returned to the notion of insight in educational practice. His discussion of what this means is clarifying. It links up with his theory of levels. Overall it makes sense. As an author he is free in the way how he presents his findings. He builds it up, from the concrete to the abstract. Wilbrink does not respect Van Hiele’s judgement, but provides no other argument than Brrrrr or the spraying with the label of Ψ or invoking the spell of the double blind randomized trial.

16. Having a hammer doesn’t make you a carpenter

Wilbrink (2012) doesn’t comment on Van Hiele’s thesis’s final chapter XVIII about the relevance of the theory of levels for epistemology. An ostrich keeps its head in the sand, where it is warm and dark, like in the womb of its egg.

Conclusion

Originally, I saw some of Ben Wilbrink’s texts on Van Hiele before, and appreciated them for the discussion and references, since there is hardly anyone else in Holland who pays attention to Van Hiele. However, Wilbrink’s reaction to Ohlsson, to the effect that Van Hiele would be wrong about the learning direction of concrete to abstract, caused me to make this evaluation above.

Wilbrink maltreats Van Hiele’s work. Wilbrink doesn’t know enough about mathematics education research (MER) to be able to write about it adequately. He misinforms the public.

I have asked Wilbrink to make adequate corrections, or otherwise specify his (reply) arguments so that I could look into those. He refuses either. This constitutes a breach in the integrity of science.

Mathematics education research (MER) not only looks at the requirements of mathematics and the didactics developed in the field itself, but also at psychology on cognition, learning and teaching in general, at pedagogy on the development of pupils and students, and at other subjects, such as physics or economics for cases when mathematics is applied, or general philosophy indeed. The former weblog text said something about neuro-psychology. Today we have a look at cognitive psychology.

“(…) the human mind also possesses the ability to override experience and adapt to changing circumstances. People do more than adapt; they instigate change and create novelty.” (cover text)

“If prior experience is a seriously fallible guide, learning cannot consist solely or even primarily of accumulating experiences, finding regularities therein and projecting those regularities onto the future. To successfully deal with thoroughgoing change, human beings need the ability to override the imperatives of experience and consider actions other than those suggested by the projection of that experience onto the situation at hand. Given the turbulent character of reality, the evolutionary strategy of relying primarily on learned rather than innate behaviors drove the human species to evolve cognitive mechanisms that override prior experience. This is the main theme of this book, so it deserves a label and an explicit statement:

The Deep Learning Hypothesis

In the course of shifting the basis for action from innate structures to acquired knowledge and skills, human beings evolved cognitive processes and mechanisms that enable them to suppress their experience and override its imperatives for action.” (page 21)

Stellan Ohlsson’s book (2011) (Source: CUP)

Definition & Reality methodology

The induction question is how one can know whether all swans are white. Even a statistical statement runs into the problem that the error is unknown. Skepticism that one cannot know anything is too simple. Economists have the question how one can make a certain general statement about the relation between taxation and unemployment.

My book DRGTPE (2000, 2005, 2011) (PDF online) (though dating from 1990, see the background papers from 1992) proposes the Definition & Reality methodology. (1) The model contains definitions that provide for certainty. Best would be logical tautologies. Lack of contrary evidence allows room for other definitions. (2) When one meets a black “swan” then it is no swan. (3) It is always possible to choose a new model. When there are so many black “swans” that it becomes interesting to do something with them, then one can define “swan2”, and proceed from there. Another example is that in one case you must prove the Pythagorean Theorem and in the other case you adopt it as a definition for the distance metric that gives you Euclidean space. The methodology allows for certainty in knowledge but of course cannot prevent surprises in empirical application or future new definitions. The methodology allows DRGTPE to present a certain analysis about a particular scheme in taxation – the tax void – that causes needless unemployment all over the OECD countries.

Karl Popper (1902-1994) was trained as a psychologist, and there met with the falsification approach by Otto Selz (1881-1943). Popper turned this into a general philosophy of science. (Perhaps Selz already thought in that direction though.) The Definition & Reality methodology is a small amendment to falsificationalism. Namely, definitions are always true. Only their relevance for a particular application is falsifiably. A criterion for a scientific theory is that it can be falsified, but for definitions the strategy is to find general applicability and reduce the risk of falsification. In below table, Pierre van Hiele presented his theory of levels of insight as a general theory of epistemology, but it is useful to highlight his original application to mathematics education, with the special property of formal proof. Because of this concept of proof, mathematics may have a higher level of insight / abstraction overall. Both mathematics and philosophy also better take mathematics education research as their natural empirical application, to avoid the risk of getting lost in abstraction.

Addendum September 7: The above assumes sensible definitions. Definitions might be logically nonsensical, see ALOE or FMNAI. When a sensible definition doesn’t apply to a particular situation, then we say that it doesn’t apply, rather than that it would be untrue or false. An example is an econometric model that consists of definitions and behavioural equations. A definition that has no relevance for the topic of discussion is not included in that particular model, but may be of use in another model.

(Un-) certainty

Definitions

Constants

Contingent

Mathematics

Euclidean space

Θ = 2π

?

Physics

Conservation of energy

Speed of light

Local gravity on Earth

Economics

Savings are income minus consumption

Institutional (e.g. annual tax code)

Behavioural equations

Mathematics education

Van Hiele levels of insight

Institutional

Student variety

To my great satisfaction, Ohlsson (2011:234) adopts basically the same approach.

“The hypothetical process that supposedly transforms particulars into abstractions is called induction and it is often claimed to operate by extracting commonalities across multiple particulars. If the first three swans you ever see are white, the idea swans are white is likely to come to mind. However, the notion of induction is riddled with problems. How are experiences grouped for the purpose of induction? That is, how does the brain know which experiences are instances of some abstraction X, before that abstraction has been learned? How many instances are needed? Which features are to be extracted? How are abstractions with no instances in human experience such as the infinite, the future and perfect justice acquired?”

Definition of abstraction

There is an issue w.r.t. the definition of abstraction though. Compare:

My definition of abstraction is leaving out aspects, see here on this weblog, and see FMNAI. My suggestion is that thought itself consist of abstractions. Abstraction depends upon experience since experience feeds brain and mind, but abstraction does not depend upon repeated experience.

Ohlsson (2011:16) takes it as identical to induction, which explains the emphasis upon experience in his title, rather taken as repetition: “Memories of individual events are not very useful in themselves, but, according to the received view, they form the raw material for further learning. By extracting the commonalities across a set of related episodic memories, we can identify the underlying regularity, a process variously referred to as abstraction, generalization or induction.” For Ohlsson, thoughts do not consists of abstractions, but of representations (models): “In the case of human cognition – or the intellect, as it would have been called in the 19th century – the relevant stuff consists of representations. Cognitive functions like seeing, remembering, thinking and deciding are implemented by processes that create, utilize and revise representations.” and “Representations are structures that refer to something (other than themselves).” (page 29)

Ohlsson has abstraction ⇔ induction (commonality). For me it is dubious whether induction really exists. The two pathways are too different to use equivalence. (i) Comparing A and B, one must first abstract from A and then abstract from B, before one may decide whether those abstractions are the same, and before one can even say that A and B share a commonality. (ii) An abstract idea like a circle might cause an “inductive” statement that all future empirical circles will tend to be round, but this isn’t really what is meant by “induction” – which is defined as the “inference” from past swans to future swans.

For me, an abstraction can be a model too, and thus would fit Ohlsson’s term representation, but the fact that he chooses abstraction ⇔ induction rather than abstraction ⇔ representation causes conceptual problems. Ohlsson’s definition of abstraction seems to hinder his understanding of the difference between concrete versus abstract as used in mathematics education research (MER).

Concrete versus abstract

Indeed, Ohlsson suggests an inversion of how people arrive at insight:

“The second contribution of the constraint-based theory is the principle that practical knowledge starts out general and becomes more specific in the course of learning. There is a long-standing tradition, with roots in the beginnings of Western philosophy, of viewing learning as moving in the opposite direction, from particulars to abstractions. [ftnt 38 e.g. to Piaget] Particulars are given in perception while abstractions are human constructions, or so the ancient story goes.” (p234)

“The fundamental principle behind these and many other cognitive theories is that knowledge moves from concrete and specific to abstract and general in the course of learning.” (Ohlsson 2011:434 that states ftnt 38)

If I understand this correctly, and combine this with the earlier argument that general knowledge is based upon induction from specific memories, then we get the following diagram. Ohlsson’s theory seems inconsistent, since the specific memories must derive from specific knowledge but also presume those. Perhaps a foetus starts with a specific memory without knowledge, and then a time loop starts with cumulation over time, like the chicken-egg problem. But this doesn’t seem to be the intention.

Trying to understand Ohlsson’s theory of knowledge

There is this statement on page 31 that I find confusing since now abstractions [inductions ?] depend upon representations, while earlier we had them derived from various memories.

“The power of cognition is greatly increased by our ability to form abstractions. Mathematical concepts like the square root of 2 and a four-dimensional sphere are not things we stumble on during a mountain hike. They do not exist except in our representations of them. The same is true of moral concepts like justice and fairness, as well as many less moral ones like fraud and greed. Without representation, we could not think with abstractions of any kind, because there is no other way for abstract entities to be available for reflection except via our representations of them. [ftnt 18]”

Ftnt 18 on page 402: “Although abstractions have interested philosophers for a long time, there is no widely accepted theory of exactly how abstractions are represented. The most developed candidate is schema theory. (…)”

My suggestion to Ohlsson is to adopt my terminology, so that thought, abstraction and representation cover the same notion. Leave induction to the philosophers, and look at statistics for empirical methods. Then eliminate representation as a superfluous word (except for representative democracy).

That said, we still must establish the process from concrete to abstract knowledge. This might be an issue of terminology too. There are some methodological principles involved however.

Wilbrink on Ohlsson

Dutch psychologist Ben Wilbrink alerted me to Ohlsson’s book – and I thank him for that. My own recent book A child wants nice and no mean numbers(CWNN) (PDF online) contains a reference to Wilbrink’s critical discussion of arithmetic in Dutch primary schools. Holland suffers under the regime of “realistic mathematics education” (RME) that originates from the Freudenthal “Head in the Clouds Realistic Mathematics” Institute (FHCRMI) in Utrecht. This FHCRMI is influential around the world, and the world should be warned about its dismal practices and results. Here is my observation that Freudenthal’s approach is a fraud.

Referring to Ohlsson, Wilbrink suggests that the “level theory by Piaget, and then include the levels by Van Hiele and Freudenthal too” (my translation) are outdated and shown wrong. This, however, is too fast. Ohlsson indeed refers to Piaget (stated ftnt 38) but Van Hiele and Freudenthal are missing. It may well be that Ohlsson missed the important insight by Van Hiele. It may explain why Ohlsson is confused about the directions between concrete and abstract.

A key difference between Van Hiele and Freudenthal

CWNN pages 101-106 discusses the main difference between Hans Freudenthal (1905-1990) and his Ph.D. student Pierre van Hiele (1909-2010). Freudenthal’s background was abstract mathematics. Van Hiele was interested from early on in education. He started from Piaget’s stages of development but rejected those. He discovered, though we may as well say defined, levels of insight, starting from the concrete to the higher abstract. Van Hiele presented this theory in his 1957 thesis – the year of Sputnik – as a general theory of knowledge, or epistemology.

Freudenthal accepted this as a thesis, but, mistook this as the difference between pure and applied mathematics. When Freudenthal noticed that his prowess in mathematics was declining, he offered himself the choice of proceeding his life with the history of mathematics or the education of mathematics. He chose the latter. Hence, he coined the phrase realistic mathematics education (RME), and elbowed Van Hiele out of the picture. As an abstract thinking mathematician, Freudenthal created an entire new reality, not caring about the empirical mindset and findings by Van Hiele. One should really read CWNN pages 101-106 for a closer discussion of this. Van Hiele’s theory on knowledge is hugely important, and one should be aware how it got snowed under.

A recent twist in the story is that David Tall (2013) rediscovered Van Hiele’s theory, but wrongly holds (see here) that Tall himself found the general value while Van Hiele had the misconception that it only applied to geometry. In itself it is fine that Tall supports the general relevance of the theory of levels.

The core confusion by Ohlsson on concrete versus abstract

The words “concrete” and “abstract” must not be used as absolutely fixed in exact meaning. This seems to be the core confusion of Ohlsson w.r.t. this terminology.

When a child plays with wooden blocks we would call this concrete, but our definition of thought is that thinking consists of abstractions, whence the meanings of the two words become blurred. The higher abstract achievement of one level will be the concrete base for the next level. The level shift towards more insight consists of compacting earlier insights. What once was called “abstract” suddenly is called “concrete”. The statement “from concrete to abstract” indicates both the general idea and a particular level shift.

Van Hiele’s theory is essentially a logical framework. It is difficult to argue with logic:

A novice will not be able to prove laws or the theorems in abstract mathematics, even informally, and may even lack the notion of proof. Having achieved formal proof may be called the highest level.

A novice will not be able to identify properties and describe their relationships. This is clearly less complex than (1), but still more complex than (3). There is no way going from (3) to (1) without passing this level.

A novice best starts with what one knows. This is not applied mathematics, as Freudenthal fraudently suggested, but concerns the development of abstractions that are available at this level. Thus, use experience, grow aware of experience, use the dimensions of text, graph, number and symbol, and develop the thoughts about these.

Van Hiele mentioned five levels, e.g. with the distinction between informal and formal deduction, but this is oriented at mathematics, and above trident seems sufficient to establish the generality of this theory of knowledge. A key insight is that words have different meanings depending upon the level of insight. There are at least three different languages spoken here.

Three minor sources of confusion are

Ohlsson’s observation that one often goes from the general to the specific is correct. Children may be vague about the distinction between “a man” and “one man”, but as grown up lawyers they will cherish it. This phenomenon is not an argument against the theory of levels. It is an argument about becoming precise. It is incorrect to hold that “one man” is more concrete and “a man” more abstract.

There appears to exist a cultural difference between on one side Germans who tend to require the general concept (All men are mortal) before they can understand the particular (Socrates is mortal), and the English (or Anglo-Saxons who departed from Germany) who tend to understand only the particular and to deny the general. This cultural difference is not necessarily epistemological.

Education concerns knowledge, skill and attitude. Ohlsson puts much emphasis on skill. Major phases then are arriving at a rough understanding and effectiveness, practicing, mastering and achieving efficiency. One can easily see this in football, but for mathematics there is the interplay with the knowledge and the levels of insight. Since Ohlsson lacks the levels of insight, his phases give only part of the issue.

Conclusion

I have looked only at parts of Ohlsson’s book, in particular above sections that allow a bit more clarity on the relevance w.r.t. Van Hiele’s theory of levels of insight. Please understand my predicament. Perhaps I read more of Ohlsson’s book later on, but this need not be soon.

In mathematics education research (MER) we obviously look at findings of cognitive psychology, but this field is large, and it is not the objective to become a cognitive psychologist oneself.

When cognitive psychologists formulate theories that include mathematical abstraction, as Ohlsson does, let them please look at the general theory on knowledge by Pierre van Hiele, for this will make it more relevant for MER.

Perhaps cognitive psychologists should blame themselves for overlooking the theory by Pierre van Hiele, but they also should blame Hans Freudenthal, and support my letter to IMU / ICMI asking to correct the issue. They may work at universities that also have departments of mathematics and sections that deal with MER, and they can ask what happened.

When there is criticism on the theory by Van Hiele, please look first at the available material. There are summary statements on the internet, but these are not enough. David Tall looked basically at one article and misread a sentence (and his misunderstanding still was inconsistent with the article). For some references on Van Hiele look here. (There is the Van Hiele page by Ben Wilbrink, but, as said, Wilbrink doesn’t understand it yet.)

Abstraction has been defined in the preceding discussion. A convenient sequel concerns what is commonly called ‘mathematical induction’. This is an instance of abstraction.

Mathematical induction has a wrong name

Mathematical induction has a wrong name. It is a boy called Sue. It is czar Putin called president. There is no induction in ‘mathematical induction’. The term is used to indicate that each natural number n has a next one, n+1. Thus for number 665 the mathematician induces 666: big surprise. And then 667 again, even a bigger surprise after 666 should be the end of the world. The second confusion is that the full name of ‘proof by mathematical induction’ is often shortened to only ‘mathematical induction’: which obscures that it concerns a method of proof only.

This method applies to the natural numbers. It actually is a deduction based upon the definition of the natural numbers. Since the natural numbers are created by numerical succession, a proper name for the method is proof by numerical succession.

Let us define the natural numbers and then establish this particular method of proof. It is assumed that you are familiar with the decimal system so that we don’t have to develop such definitions. It is also assumed that zero is a cardinal number.

Definition of the natural numbers

A finite sequence of natural numbers is N[5] = {0, 1, 2, 3, 4, 5}. Since we can imagine such sequences for any number, there arises the following distinction given by Aristotle. He called it the difference between potential and actual infinity.

(1) Potential infinity: N[n] = {0, 1, 2, 3, …., n}. This reflects the human ability to count. (1a) It uses the successor function (“+1”): s[n] = n + 1. For each n there is a n+1. The successor function is a primitive notion that cannot be defined. You get it or you don’t get it. As a formula we can ‘define’ it by writing ‘For each n there is a n+1′, but this is not really a definition but rather the establishment of a convention how to denote it. (1b) Numerical succession might actually be limited to a finite number, say for a window of a small calculator that allows for 6 digits: 0 ≤ n ≤ 999,999. The crux of N[n] however is that n can be chosen and re-chosen at will. For each N[n] we can choose a N[n+1].

(2) Actual infinity: N = {0, 1, 2, 3, …}. This reflects the human ability to give a name to some totality. Here the name is ‘the natural numbers’.

The potential infinite deals with finite lists. Each list has a finite length. The distinctive property of these lists is that for each such number one can find a longer list. But they are all finite. It is an entirely different situation to shift to the actual infinite, in which there is a single list that contains all natural numbers.

There need be no doubt about the ‘existence’ of the natural numbers. The notion in our minds suffices. However, our mental image may also be a model for reality. If the universe is finite, then it will not contain an infinite line, and there cannot be a calculator with a window of infinite length. But, on every yardstick in the range [0, 1] we have all 1H, 2H, 3H, ….. PM. We denote nH = 1 / n, to be pronounced as per-n, see the earlier discussion on nH.

The relation between potential and actual infinity

The shift from N[n] to N is an instance of abstraction. N[n] is a completed whole but with a need to build it, with a process of repetition. N ‘leaves out’ that one is caught in some process of repetition, while there still is a completed whole. Let us use a separate symbol @ for the particular kind or instance of abstraction that occurs in the shift from (1) to (2).

(3) N[n] @ N. This records that (1) and (2) are related in their concepts and notations. In the potential form for each n there is a n+1. In the actual form there is a conceptual switch to some totality, caught in the label N.

Since we already defined (1) and (2) to our satisfaction, (3) is entirely derivative and does not require an additional definition. It merely puts (1) and (2) next to each other, while the symbol ‘@’ indicates the change in perspective from the potential to the actual infinite.

(There might be a link to the notion of ‘taking a limit’ but it is better to leave the word ‘limit’ to its well-defined uses and take ‘@’ as capturing above instance of abstraction.)

Proof by numerical succession

The method of proof by numerical succession follows the definition of the natural numbers.

Definition: Let there be a property P[n] that depends upon natural number n. The property can be established – or become a theorem – for all natural numbers n ≥ m, by the following method of proof, called the method by numerical succession: (i) show that P[m] holds, (ii) show that P[n-1] ⇒ P[n]. (The validity of the proof depends upon whether these two steps have been taken well of course.)

When m = 0 then the property might hold for all natural numbers. The second step copies the definition of N: If n-1 ∈ N and P[n-1], then n ∈ N and then it must be shown that P[n]: if it is to hold that P[n] for all n ∈ N.

PM 1 below contains an example that uses a more conventional notation of going from n to n+1.

The definition of the method of proof doesn’t state this explictly: In the background there always is (N[n] @ N) w.r.t. the fundamental distinction between the finite N[n] and the infinite N. Conceivably we could formulate a method for N[n] separately that emphasizes the finitary view but there is no need for that here.

Conclusions

(1) A prime instance of abstraction is the relation N[n] @ N, i.e. the shift from the potential to the actual infinity of natural numbers.

(2) The method of ‘proof by numerical succession’ is a deductive method based upon the definition of the natural numbers.

(3) ‘Proof by numerical succession’ is a proper name, for what confusingly is called ‘proof by mathematical induction’.

(4) Without further discussion: There is no ‘unreasonable effectiveness’ in the creation of the infinity of the natural numbers and the method of proof by numerical succession, and thus neither in the application to the natural sciences, even when the natural sciences would only know about a finite number (say number of atoms in the universe).

PM 2. Background theory

PM 3. Rejection of alternative names

The name ‘mathematical succession’ can be rejected since we are dealing with numbers while mathematics is wider. The name ‘natural succession’ can be rejected since it doesn’t refer to mathematics – consider for example the natural succession to Putin. The name ‘succession for the natural numbers’ might also be considered but ‘numerical succession’ is shorter and on the mark too.

PM 4. Wikipedia acrobatics

Earlier we diagnosed that wikipedia is being terrorized by students from MIT who copy their math books without considering didactics. The wiki team seems to grow aware of the challenge and is developing a ‘simple wiki’ now. Check the standard articleon mathematical induction and the simple article. The next steps for the wiki team are: to establish the distinction between easy and their notion of simplicity, then reduce the standard wiki into an easy one, and subsequently ask the MIT students to do both their copying and their experiments on simplicity at this ‘simple wiki’.

Thinking depends upon abstraction. Let Isaac Newton observe an apple falling from a tree. The apple and the tree are concrete objects. The observation consists of processes in Newton’s mind. The processes differ from the concrete objects and leave out a wide range of aspects. This is the definition of abstraction: to leave out aspects. Perhaps nature “thinks” by means of the concrete objects, but a mind necessarily must omit details and can only deal with such abstractions. For example, when Newton suddenly is hit by the idea of the universal law of gravity, then this still is an idea in his mind, and not the real gravity that the apple – and he himself – are subjected to.

Newton discovers the universal law of gravity, (c) marcelgotlib.com

Edward Frenkel’s reference to Eugene Wigner

There is this quote:

“The concepts that Yang and Mills used to describe forces of nature appeared in mathematics earlier because they were natural also within the paradigm of geometry that mathematicians were developing following the inner logic of the subject. This is a great example of what another Nobel Prize-winner, physicist Eugene Wigner, called the “unreasonable effectiveness of mathematics in the natural sciences.” [ref] Though scientists have been exploiting this “effectiveness” for centuries, its roots are still poorly understood. Mathematical truths seem to exist objectively and independently of both the physical world and the human brain. There is no doubt that the links between the world of mathematical ideas, physical reality, and consciousness are profound and need to be further explored. (We will talk more about this in Chapter 18.)” (Edward Frenkel, “Love & Math”, 2013, p 202, my emphasis)

Hopefully you spot the confusion. Frenkel is an abstract thinking mathematician with some experience in science – e.g. with a patent – but apparently without having understood the philosophy of science. This weblog has already discussed some of his views, see here, especially his confusion about mathematics education while he hasn’t studied the empirical science of didactics. It is a chilling horror to hear him lecture about how math should be taught and then see the audience listening in rapture because they think that his mathematical brilliance will certainly also generate truth in this domain.

Eugene Wigner’s error – see the paper below – is to forget that abstraction still is based upon reality. When reality consists of {A, B, C, …, Z} and you abstract from this reality by looking only at A and leaving out {B, C, …., Z} then it should not surprise you that A still applies to reality since it has been taken from there.

Mathematical ideas have a perfection that doesn’t seem to exist in concrete form in reality. A circle is perfectly round in a manner that a machine likely cannot reproduce – and how would we check ? If the universe has limited size then it cannot contain a line, which is infinite in both directions. Both examples however are or depend upon abstractions from reality.

Since mathematics consists of abstractions, we should not be surprised when its concepts don’t fully apply to reality, and neither should we be surprised when some applications do. That is, there is no surprise in terms of philosophy. In practice we can be surprised, but this is only because we are mere human.

“It seems to me that the question that Wigner is asking is “Why is mathematics so much more effective in physics (which is what he means by ‘natural sciences’) than in most other studies?” Physics textbooks are full of formulas; these comprise a large fraction of what the field is, and have great predictive power. Textbooks on invertebrate biology have few mathematical formulas, and they comprise only a small part of the field. Textbooks on comparative literature mostly have no formulas. So an answer to Wigner’s question would have to say something about what it is about physics _specifically_ that lends itself to mathematimization; merely appealing to the human desire for abstraction doesn’t explain why physics is different from these other fields.
I have no idea what an answer to Wigner’s question could possibly look like. My feeling is that it is better viewed as an expression of wonderment than as an actual question that expects an answer.” (Comment made anonymous, January 9 2015)

I don’t agree with this comment. In my reading, Wigner really poses the fundamental philosophical question, and not a question about a difference in degree between physics and literature. The philosophical question is about the relation between abstraction and reality. And that question is answered by reminding about the definition of abstraction.

I can agree that physics seems to be more mathematical in degree than literature, i.e. when we adopt the common notions about mathematics. This obviously has to do with measurement. Use a lower arm’s length, call this an “ell“, and proceed from there. Physics only has taken the lead – and thus has also the drawbacks of having a lead (Jan Romein’s law). Literature however also exists in the mind, and thus also depends upon abstractions. Over time these abstractions might be used for a new area of mathematics. Mathematics is the study of patterns. Patterns in literature would only be more complex than those in physics – and still so inaccessible that we call them ‘subjective’.

For example, the patterns in Gotlib’s comic literature about Newton & his apple might be more complex than the patterns in the physics of Newton & his apple, as described by his universal law of gravity. All these remain abstract and differ from the concrete Newton & his apple.

There is no “unreasonable effectiveness” in that Gotlib’s comic makes us smile.